Abstract
One approach to achieving correct finite element assembly is to ensure that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet. Rognes et al. have shown how to achieve this for any mesh composed of simplex elements, and deal.II contains a serial algorithm for constructing a consistent orientation of any quadrilateral mesh of an orientable manifold. The core contribution of this paper is the extension of this algorithm for distributed memory parallel computers, which facilitates its seamless application as part of a parallel simulation system. Furthermore, our analysis establishes a link between the well-known Union-Find algorithm and the construction of a consistent orientation of a quadrilateral mesh. As a result, existing work on the parallelization of the Union-Find algorithm can be easily adapted to construct further parallel algorithms for mesh orientations.
Highlights
A characteristic of the finite element method for the solution of partial differential equations is that the representation of functions over the domain can be chosen from a wide range of function spaces
We used 6 different meshes to evaluate the performance of the parallel algorithm described in section 5: s square: structured grid on a square domain s sphere: cubed sphere mesh u square: unstructured mesh on a square domain u sphere: unstructured mesh on the surface of a sphere t10, t11: unstructured meshes with nonuniform resolution The first two were generated using Firedrake utility functions, and the other four were generated with Gmsh [10], version 2.8.3. t10 and t11 are examples from the Gmsh tutorial
We have proposed a distributed mesh parallel algorithm, extending the serial quadrilateral edge orientation algorithm in deal.II
Summary
A characteristic of the finite element method for the solution of partial differential equations is that the representation of functions over the domain can be chosen from a wide range of function spaces. Every finite element implementation which supports facet integrals, elements of polynomial degree greater than one, or H(div) or H(curl) conforming elements must somehow ensure that adjacent cells agree on the orientation of the intervening facet. This can either be achieved by explicitly recording orientation information and exploiting this in the local and/or global assembly operations, or it can be achieved by ensuring, in a sense which we will later make formal, that the local orientation of facets relative to each cell in the mesh is consistent with the global orientation of that facet
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